The simple answer is yes. If you maintained the same angle with respect to space you would be pitching up relative to earth and that would initially cause a gain in altitude which is not desirable in the long term.
The proper answer is that it happens so slowly you can’t feel it.
At a typical airliner speed of 450 knots (nautical miles per hour) you would be doing 7.5 nautical miles per minute (450/60).
If you fly at that speed you will cover the equivalent of 1 degree of latitude (60 nautical miles) every 8 minutes (60/7.5).
So at 450 knots you would have to pitch down 1/8th of a degree every minute.
Blowing this up to a bigger scale, it’s 10,800 miles from pole to pole and that would mean 180 degrees of pitch change, @450kts that would take 24 hours (10800/450) or 1440 minutes. 180 degrees / 1440 minutes = 0.125 degrees per minute or 1/8th of a degree.
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